The quantizations of Teichmuller space I have seen are via special coordinates (e.g. the paper of Chekhov and Fock) or conformal blocks. Does one get an equivalent quantization by geometric quantization? For example the Weil-Peterson metric is Kahler. Is it known, one way or the other, if doing Kahler quantization give the same thing? What about a different polarization?

The quantizations of Teichmuller space I have seen are via special coordinates (e.g. the paper of Chekhov and Fock) or conformal blocks. Does one get an equivalent quantization by geometric quantization? For example the Weil-Peterson metric is Kahler. Is it known, one way or the other, if doing Kahler quantization gives the same thing? What about a different polarization?